# Resolvent estimates for spacetimes bounded by Killing horizons

## TL;DR

This paper proves that on certain stationary spacetimes with Killing horizons, the wave equation's resolvent grows at most exponentially with frequency, leading to small resonance-free regions and logarithmic energy decay of solutions.

## Contribution

It establishes exponential bounds on the resolvent growth for wave equations on spacetimes with Killing horizons without assuming trapped set conditions.

## Key findings

- Resolvent grows at most exponentially with frequency
- Existence of exponentially small resonance-free regions
- Solutions exhibit logarithmic energy decay

## Abstract

We show that the resolvent grows at most exponentially with frequency for the wave equation on a class of stationary spacetimes which are bounded by non-degenerate Killing horizons, without any assumptions on the trapped set. Correspondingly, there exists an exponentially small resonance-free region, and solutions of the Cauchy problem exhibit logarithmic energy decay.